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Properties

Label 1-149-149.140-r0-0-0
Degree $1$
Conductor $149$
Sign $-0.568 - 0.822i$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.450 − 0.892i)2-s + (0.210 − 0.977i)3-s + (−0.594 + 0.803i)4-s + (−0.127 + 0.991i)5-s + (−0.967 + 0.251i)6-s + (0.942 − 0.333i)7-s + (0.985 + 0.169i)8-s + (−0.911 − 0.411i)9-s + (0.942 − 0.333i)10-s + (−0.594 − 0.803i)11-s + (0.660 + 0.750i)12-s + (0.372 − 0.927i)13-s + (−0.721 − 0.691i)14-s + (0.942 + 0.333i)15-s + (−0.292 − 0.956i)16-s + (0.210 − 0.977i)17-s + ⋯
L(s)  = 1  + (−0.450 − 0.892i)2-s + (0.210 − 0.977i)3-s + (−0.594 + 0.803i)4-s + (−0.127 + 0.991i)5-s + (−0.967 + 0.251i)6-s + (0.942 − 0.333i)7-s + (0.985 + 0.169i)8-s + (−0.911 − 0.411i)9-s + (0.942 − 0.333i)10-s + (−0.594 − 0.803i)11-s + (0.660 + 0.750i)12-s + (0.372 − 0.927i)13-s + (−0.721 − 0.691i)14-s + (0.942 + 0.333i)15-s + (−0.292 − 0.956i)16-s + (0.210 − 0.977i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $-0.568 - 0.822i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ -0.568 - 0.822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4094946040 - 0.7810591753i\)
\(L(\frac12)\) \(\approx\) \(0.4094946040 - 0.7810591753i\)
\(L(1)\) \(\approx\) \(0.6742814158 - 0.5649059584i\)
\(L(1)\) \(\approx\) \(0.6742814158 - 0.5649059584i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.450 - 0.892i)T \)
3 \( 1 + (0.210 - 0.977i)T \)
5 \( 1 + (-0.127 + 0.991i)T \)
7 \( 1 + (0.942 - 0.333i)T \)
11 \( 1 + (-0.594 - 0.803i)T \)
13 \( 1 + (0.372 - 0.927i)T \)
17 \( 1 + (0.210 - 0.977i)T \)
19 \( 1 + (0.0424 - 0.999i)T \)
23 \( 1 + (0.372 + 0.927i)T \)
29 \( 1 + (0.873 + 0.487i)T \)
31 \( 1 + (0.372 + 0.927i)T \)
37 \( 1 + (-0.594 - 0.803i)T \)
41 \( 1 + (-0.292 - 0.956i)T \)
43 \( 1 + (0.524 - 0.851i)T \)
47 \( 1 + (0.0424 + 0.999i)T \)
53 \( 1 + (0.524 + 0.851i)T \)
59 \( 1 + (-0.996 - 0.0848i)T \)
61 \( 1 + (-0.450 - 0.892i)T \)
67 \( 1 + (-0.721 + 0.691i)T \)
71 \( 1 + (-0.127 + 0.991i)T \)
73 \( 1 + (0.778 + 0.628i)T \)
79 \( 1 + (0.778 - 0.628i)T \)
83 \( 1 + (0.778 + 0.628i)T \)
89 \( 1 + (-0.828 + 0.559i)T \)
97 \( 1 + (0.524 + 0.851i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.246342737857245819606675402719, −27.44228727224615240671256593290, −26.52346975981810354118024796404, −25.592596491710236943036660349566, −24.66465666564791426569959950987, −23.730153613089854919045136200, −22.8070195969385540185187286447, −21.28091249894951968276381678131, −20.72976304719056720328322548989, −19.5159440137908940741914417384, −18.26952166483402308240987634708, −17.07355341647952082165804025526, −16.48599466517277748848878417370, −15.375987234030167199880173271370, −14.725560405887394568474842774462, −13.5616896056751616113618318397, −12.01360051063029805944430977692, −10.58324018256173473174149370949, −9.58142651741166737570303407051, −8.491030512247811283537145800072, −7.98064755433774098404082949899, −6.04422818993475953819663914103, −4.86638020672221305129700388630, −4.27888789023979932651807465425, −1.75930169510483242894584004392, 0.95563730997590794887549818785, 2.53587201102780398193324160833, 3.34995739476853462982494678556, 5.30656362781146801138337042828, 7.12300108197102295430357125047, 7.852982191837672399643080232, 8.891625010455282047386339774769, 10.62039461345976541685075123492, 11.16859373461845904138566665463, 12.2301221010641435167635696492, 13.631495458803924209060457078861, 14.031740544187692289952081377030, 15.64461911041665187340603350493, 17.47940783160269271280410228411, 17.94787547110623119295292180806, 18.795025523311457601170730707360, 19.67534961906932169957996692661, 20.66671828785624273222660909023, 21.68619031825318931860996219541, 22.91858058245785406843804298920, 23.65648108933629380796888621486, 25.02775687591252633732021790887, 26.00324343560794319235445263416, 26.906065735029319872324808928364, 27.706049879316189201342935079662

Graph of the $Z$-function along the critical line